An automated palmprint recognition system.

Conference Proceeding: Efficient joint 2D and 3D palmprint matching with alignment refinement.
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ABSTRACT: Palmprint verification is a relatively new but promising personal authentication technique for its high accuracy and fast matching speed. Two dimensional (2D) palmprint recognition has been well studied in the past decade, and recently three dimensional (3D) palmprint recognition techniques were also proposed. The 2D and 3D palmprint data can be captured simultaneously and they provide different and complementary information. 3D palmprint contains the depth information of the palm surface, while 2D palmprint contains plenty of textures. How to efficiently extract and fuse the 2D and 3D palmprint features to improve the recognition performance is a critical issue for practical palmprint systems. In this paper, an efficient joint 2D and 3D palmprint matching scheme is proposed. The principal line features and palm shape features are extracted and used to accurately align the palmprint, and a couple of matching rules are defined to efficiently use the 2D and 3D features for recognition. The experiments on a 2D+3D palmprint database which contains 8000 samples show that the proposed scheme can greatly improve the performance of palmprint verification.The TwentyThird IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2010, San Francisco, CA, USA, 1318 June 2010; 01/2010  SourceAvailable from: Phalguni Gupta
Article: Verification system robust to occlusion using loworder Zernike moments of palmprint subimages.
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ABSTRACT: This paper proposes a palmprint based verification system which uses loworder Zernike moments of palmprint subimages. Euclidean distance is used to match the Zernike moments of corresponding subimages of query and enrolled palmprints. These matching scores of subimages are fused using a weighted fusion strategy. The proposed system can also classify the subimage of palmprint into nonoccluded or occluded region and verify user with the help of nonoccluded regions. So it is robust to occlusion. The palmprint is extracted from the acquired hand image using a low cost flat bed scanner. A palmprint extraction procedure which is robust to hand translation and rotation on the scanner has been proposed. The system is tested on IITK, PolyU and CASIA databases of size 549, 5239 and 7752 hand images respectively. It performs with accuracy of more than 98%, and FAR, FRR less than 2% for all the databases.Telecommunication Systems 01/2011; 47:275290. · 1.03 Impact Factor  SourceAvailable from: Aythami Morales[show abstract] [hide abstract]
ABSTRACT: Personal recognition through handbased biometrics has attracted the interest of many researchers in the last twenty years. A significant number of proposals based on different procedures and acquisition devices have been published in the literature. However, comparisons between devices and their interoperability have not been thoroughly studied. This paper tries to fill this gap by proposing procedures to improve the interoperability among different hand biometric schemes. The experiments were conducted on a database made up of 8,320 hand images acquired from six different hand biometric schemes, including a flat scanner, webcams at different wavelengths, high quality cameras, and contactless devices. Acquisitions on both sides of the hand were included. Our experiment includes four feature extraction methods which determine the best performance among the different scenarios for two of the most popular hand biometrics: hand shape and palm print. We propose smoothing techniques at the image and feature levels to reduce interdevice variability. Results suggest that comparative hand shape offers better performance in terms of interoperability than palm prints, but palm prints can be more effective when using similar sensors.Sensors 01/2012; 12(2):135282. · 1.95 Impact Factor
Page 1
An automated palmprint recognition system
Tee Connie*, Andrew Teoh Beng Jin, Michael Goh Kah Ong, David Ngo Chek Ling
Faculty of Information Science and Technology, Multimedia University, Jalan Ayer Keroh Lama, 75450 Melaka, Malaysia
Received 13 March 2004; received in revised form 10 November 2004; accepted 14 January 2005
Abstract
Recently, biometric palmprint has received wide attention from researchers. It is wellknown for several advantages such as stable line
features, lowresolution imaging, lowcost capturing device, and userfriendly. In this paper, an automated scannerbased palmprint
recognition system is proposed. The system automatically captures and aligns the palmprint images for further processing. Several linear
subspace projection techniques have been tested and compared. In specific, we focus on principal component analysis (PCA), fisher
discriminant analysis (FDA) and independent component analysis (ICA). In order to analyze the palmprint images in multiresolutionmulti
frequency representation, wavelet transformation is also adopted. The images are decomposed into different frequency subbands and the best
performing subband is selected for further processing. Experimental result shows that application of FDA on wavelet subband is able to yield
both FAR and FRR as low as 1.356 and 1.492% using our palmprint database.
q 2005 Elsevier B.V. All rights reserved.
Keywords: Biometric; Palmprint recognition; Palmprint preprocessing; Subspace projection methods; Similarity matching
1. Introduction
Recently, a new biometric feature based on palmprint has
been introduced. Palmprint recognition refers to the process
of determining whether two palmprints are from the same
person based on line patterns of the palm. Palmprint is
referred to the principal lines, wrinkles and ridges appear on
the palm, as showed in Fig. 1. There are three principal lines
on a typical palm, named as heart line, head line and life
line, respectively. These lines are clear and they hardly
change throughout the life of a person. Wrinkles are lines
that are thinner than the principal lines and are more
irregular. The lines other than principal lines, as well as
wrinkles, are known as ridges, and they exist all over the
palm.
Palmprint serves as a reliable human identifier because
the print patterns are not duplicated in other people, even
in monozygotic twins. More importantly, the details of
these patterns are permanent. The rich structures of the
palmprint offer plenty of useful information for recog
nition. There are two popular approaches to palmprint
recognition. The first approach is based on the palmprint
statistical features while the other on structural features.
For statistical based palmprint recognition approach, the
works that appear in the literature include eigenpalm [1],
fisherpalms [2], Gabor filters [3], Fourier Transform [4],
and local texture energy [5].
Another important feature extraction approach is to
extract structural information, like principal lines and
creases, from the palm for recognition. Funada et al. [6]
devised a minutiae extraction method for palmprints. This
idea was inspired by the fact that palmprint also contains
minutiae like fingerprints. Zhang and Shu [7] determined
the datum points derived from the principal lines using the
directional projection algorithm. These datum points were
location and rotational invariant due to the stability of the
principal lines. Unlike the work proposed by [7], Duta et al.
[8] did not explicitly extract palm lines, but used only
isolated points that lie along palm lines as they deduced that
feature point connectivity was not essential for the matching
purposes. As opposed to the work by [6], Chen et al. [4]
recognized palmprint by using creases. Their work was
motivated by the finding that some crease patterns are
related to some diseases of people. Another structural based
method used by Wu et al. [10] was to implement fuzzy
02628856/$  see front matter q 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.imavis.2005.01.002
Image and Vision Computing 23 (2005) 501–515
www.elsevier.com/locate/imavis
* Corresponding author. Tel.: C60 252 3611.
Email address: tee.connie@mmu.edu.my (T. Connie).
Page 2
directional element energy feature (FEDDF) which origi
nated from the idea of a Chinese character recognition
method called directional element feature (DEF). On the
other hand, Han et al. [11] performed Sobel’s and
morphological operations to extract palmprint structural
features from the region of interest (ROI).
In the first statistical features based palmprint recognition
approach, the palmprint image is treated as a whole for
extraction, representation and comparison. Thus, the
recognition process is straightforward. However, as abun
dant textural details are ignored, the natural and structural
information of the palmprint cannot be characterized. On
the other hand, structural approach can represent the
palmprint structural features clearly. Besides, image with
lower quality can be used for structural approach as lines
can be detected under lowresolution. However, this method
is restricted by the complication in determining the
primitives and placements of the line structures, and usually
more computational power is required to match the line
segments with the templates stored in the database. Each
approach demonstrates its strengths and weaknesses, and
the choice depends on the temperament of application:
operational mode, processing speed, memory storage and
quality of the image acquired.
In addition to the feature selection process, image
capturingmethodisanotherimportantfactortobeevaluated.
The palmprint recognition methods proposed by [4–10]
utilized inked palmprint images. These approaches are able
to provide highresolution images and are suitable for
methods which require fine resolution images to extract
lines, datum points and minutiae features. However, these
methods are not suitable for online security systems as two
steps are required to be performed: ink the palmprint images
on papers and then scan them to obtain digital images. Some
recent works demonstrated by [3,10,12] used CCD based
digital camera to capture palmprint images. The digital
images acquired could be directly fed into computer for
computation. Another approach proposed by [11] used
scanner as the acquiring device. The advantage of scanner
is that it is equipped with a flat glass that enables the users to
flatten their palm properly on the glass to reduce bended
ridges and wrinkle errors. Some authors like [4,10] fixed
some guidance pegs on the sensor’s platform to limit the
palm’sshiftandrotation.Someuserswillfeeluncomfortable
when their hands images are acquired. In addition, this
approach requires additional pegremoval algorithm to
remove the pegs from the hand image. Works introduced
by [12] do not use fixed pegs to increase flexibility and user
friendliness of the system.
In this paper, an automated pegfree scannerbased
palmprint recognition system is proposed. Two novel
components are contained in the proposed system. First, a
preprocessing module that automatically aligns palmprint
images from pegfree sensor is developed. This module
segments hand image from the background and extracts the
center region of the palm for recognition. Second,
systematic comparison and analysis of three types of
subspace projection techniques, namely principal com
ponent analysis, fisher discriminant analysis and indepen
dent component analysis, using a standard palmprint
database is presented. In order to analyze palmprint images
in multiresolutionmultifrequency representation, the
wavelet transformation is also adopted.
In the next section, the overview of the proposed
palmprint recognition system is provided and each of the
system’s components is discussed in details. Section 3
presents the experiment setup, as well as the results of this
research. In Section 4, we make some concluding remarks.
Finally, the review of PCA, FDA, ICA and Wavelet
Transform theories are provided in Appendix A for the
convenience of readers unfamiliar with these techniques.
2. Overview of system architecture
The proposed system is divided into two phases, namely
the enrollment and verification phase, as shown in Fig. 2.
The important tasks contain in the system include the
preprocessing, feature extraction as well as feature
matching. In the preprocessing stage, the alignment and
orientation of the hand images are corrected for use in the
successive tasks. In the feature extraction stage, the most
discriminating features from the palms are extracted for
representation, and finally in the feature matching stage
comparison is performed and decision is made whether two
palmprint features are from the same person. The details of
each of these components are discussed in the subsequent
sections.
2.1. Preprocessing
In this system, no guidance pegs are fixed on the
scanner’s platform and the users are allowed to place their
hands freely on the platform of the scanner when scanned.
Thus, palmprint images with different sizes, shifts and
rotations are produced. Therefore, a preprocessing algor
ithm has been developed to correct the orientation of
the images and also convert the palmprints into same size
Fig. 1. The line patterns on the palmprint. The three principal lines on a
typical palm: 1heart line, 2head line and 3life line.
T. Connie et al. / Image and Vision Computing 23 (2005) 501–515502
Page 3
images. Successful preprocessing measure can provide the
foundation for both feature extraction and matching.
Before alignment and orientation are performed on the
palmprint, a smaller region from the center of the palm,
called region of interest (ROI), is automatically extracted.
The ROI is defined in square shape and it contains sufficient
information to represent the palmprint for further proces
sing. Fig. 3 depicts the appearance of ROI of a palm.
We applied the salientpoint detection algorithm pro
posed by Goh et al. [13] to obtain the three crucial points, v1,
v2and v3as shown in Fig. 3, used to locate the ROI. First, an
image thresholding technique is applied to segment the hand
image from the background. The proposed technique can
also detect fingernails and rings by analyzing the skin color
of the hand. The hand image acquired is in 256RGB colors
with stable background in grey. The background can be
segmented based on the values of the image’s color
component r, g, and b which represent red, green and
blue, respectively. The image thresholding technique
proposed is shown in Eq. (1):
C1ðu;vÞ Z
1
jrðu;vÞKbðu;vÞj!T
otherwise0
(
(1)
Eq. (1) is repeated for setting, jr(u,v)Kg(u,v)j, yielding
C2(u,v) and jb(u,v)Kg(u,v)j, C3(u,v). The threshold value T
is set to 50 to filter all the grey level color to white and other
color to black. The resultant image of binary pixel C1, C2
and C3are ANDed to obtain the binary image, I:
I Z
X
After that, contour of the hand shape is obtained by using
eight neighborhood border tracing algorithm [14]. The
process starts by scanning the pixels of the binary image
from the bottomleft to the right. When the first black pixel
is detected the border tracing algorithm is initiated to trace
the border of the hand in clockwise directions. During the
border tracing process, all the coordinates of the border
pixels were recorded in order to represent the signature of
the hand, f(i) where i is the array index. The hand signature
h
vZ1
X
w
uZ1
h
3
iZ1Ciðu;vÞ
(2)
is blocked into nonoverlapping frames of 10 samples, f(i).
Every frame is checked for existence of stationary points
and in this way the valleys of the fingers, v1, v2and v3could
be pinpointed. Based on the information of these three
crucial points, the outline of the ROI could be obtained as
follows:
1. The two valleys beside the middle finger, v1, v2, are
connected to form a reference line.
2. The reference line is extended to intersect the rightedge
of the hand.
3. The intersection point obtained from step (2) is used to
find the midpoint, m1, based on the midpoint formula.
4. Steps (1) to (3) are repeated to find the other midpoint,
m2, by using the valleys v2, v3.
5. The two midpoints, m1and m2, are connected to form the
base line to obtain the ROI.
6. Based on the principal of geometrical square where all
the four edges having equal length, the other two points,
Fig. 2. Block diagram of the proposed palmprint verification system.
Fig. 3. Outline of the region of interest (ROI) from the palm.
T. Connie et al. / Image and Vision Computing 23 (2005) 501–515503
Page 4
m3and m4, needed to form the square outline of the ROI
can be obtained (refer Fig. 3).
Fig. 4 shows some examples of the ROIs extracted from
different individuals, obtained from both the right and left
palms.
There are some variances in the locations of the base
points, m1and m2, used to obtain the ROI. This variance is
caused by the different stretching degree of the hand.
Experimental statistic shows that the average standard
deviation of the location of the base point is approximately
2.462 pixels. However, the variance in locations of the base
points does not cause much effect to the feature extraction
process, as it only affects the capturing size of the outline of
the ROI. Most of the information significant for the
recognition task lies in the center of the ROI, thus small
variationinthelocationofthebasepointswillnotjeopardize
thesystem’sperformance.Infact,experimentalresultshows
that the system performs well using these ROI features.
From Fig. 4, it can be observed that the ROI have
different sizes, due to the varying palm’s size. Usually men
have larger palms’ sizes than the women. For example the
ROI of a man shown in Fig. 4(d) is larger in size than ROI of
a lady shown in Fig. 4(c). Besides the differences in size, all
the ROIs lie in various directions. Due to these incon
sistencies, the preprocessing job is performed to align all the
ROIs into the same location in their images.
First, the images are rotated to the rightangle position by
using Yaxis as the rotationreference axis. The next step is
to convert the RGB ROI into grayscale image. After that, as
the sizes of the ROIs vary from hand to hand (depending on
the sizes of the palms), they are resized to 150!150 pixels
images by using bicubic interpolation.
The last procedure in the preprocessing stage is to
normalize the palmprint images in order to smoothen
the noise and lighting effect. The normalization method
deployed in this research follows the discussion by Shi et al.
[15]. Let P(x, y) represents the pixel value at the coordinate
(x, y), m and n be the image mean and variance, respectively.
The normalized image is computed by using the operation
below:
P0ðx;yÞZ
mtCb ifPðx;yÞOm
mtKb otherwise
wherebZ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
ntfPðx;yÞKmg2
s
8
:
<
(3)
where mtand ntare the preset values for mean and variance
for the image. In this experiment, the value of mtand ntwere
set to 10, respectively. Fig. 5(b) depicts the palmprint image
after the normalization process.
2.2. Palmprint feature extractions
After the wellaligned ROIs are obtained from the pre
processing stage, we extract important features from
Fig. 4. ROIs obtained from different individuals. They have different sizes and rotations. (a), (b), (c) and (d) depicts ROIs from the right palms from four
individuals, while (e), (f), (g) and (h) are ROIs from the left palms from another four individuals.
Fig. 5. Extracted ROI from the palm. (a) Palmprint image before
normalization. (b) Palmprint after normalization.
T. Connie et al. / Image and Vision Computing 23 (2005) 501–515504
Page 5
the image for recognition task. As discussed in Section 1,
there are many approaches to achieve this purpose. In this
paper, three subspace projection techniques are experimen
ted and compared. In particular, we use principal component
analysis (PCA), fisher discriminant analysis (FDA) and
independent component analysis (ICA). The subspace
projection technique is performed as a twostep process of
constructing the subspace basis followed by projecting the
palmprint images into the compressed subspace. New test
images are then projected into the same subspace for image
matching. It is computationally more efficient to perform
image matching in subspaces as the dimensions have been
reduced significantly. For example, image with 22,
500 pixels (150!150) might be projected into a subspace
with only 20–60 dimensions.
2.2.1. Principal component analysis
PCA has been widely used for dimensionality reduction
in computer vision [1,16,17]. It finds a set of orthogonal
basis vectors which describe the major variations among the
training images, and with minimum reconstruction mean
square error. This is useful as it helps to decrease the
dimensions used to describe the set of images and also scale
each variable according to its relative importance in
describing the observation. The eigen bases generated
from the set of palmprint images are shown in Fig. 6(a).
As these bases have the same dimension as the original
images and are like palmprint in appearance, they are also
called eigenpalms.
2.2.2. Fisher discriminant analysis
The successful implementation of PCA in various
recognition tasks popularized the idea of matching
images in the compressed subspaces FDA is another
popular subspace projection technique which computes a
subspace that best discriminates among classes. It is
different from PCA in the aspect that it deals directly
with class separation while PCA treats images in its
entirety without considering the underlying class struc
ture. The bases generated using FDA are also known as
fisherpalms. Some appearances of fisherpalms are
depicted in Fig. 6(b).
2.2.3. Independent component analysis
While both PCA and FDA only impose independence
only up to the second order, there is also a lot of interest to
decorrelate higher order statistics from the training images
ICA is one such approach that computes the basis
components that are statistically independent or as inde
pendent as possible [18]. ICA is originally used to solve
blind source separation (BSS) problem. When applied in
palmprint recognition, the palmprint images are considered
as the mixture of an unknown set of statistically independent
source images by an unknown mixing matrix. A separating
matrix is learnt by ICA to recover a set of statistically
independent basis images. The bases generated are spatially
localized in various portions in the palmprint image, as
shown in Fig. 6(c).
Fig. 6. The first five bases generated by (a) PCA (b) FDA (c) ICA.
T. Connie et al. / Image and Vision Computing 23 (2005) 501–515505
Page 6
2.2.4. Wavelet decomposition
Multiresolution analysis of the images is performed by
using wavelet decomposition In this paper, Wavelet
Transformation is integrated into the feature extractors as
follows:
1. Decompose the palmprint image by using different
families of wavelet.
2. Retain the lowfrequency subband of the approximation
coefficients.
3. Feed the reduced images into {PCAjFDAjICA}
computation.
According to wavelet theory, it is generally found that
most of the energy content is concentrated in the low
frequency subband, as compared to higher frequency
subbands. Low frequency subband is the smoothed version
of the original image and it helps to reduce the influence of
noise on one hand, and on the other hand preserves the local
edges well which helps to capture the features that is
insensitive to small distortion. On the other hand, the higher
frequency subbabds only contain low energy content and
their high pass feature tends to enhance the edges detail,
including noise and shape distortion.
WT is selected upon other filtering designs as it can
decompose the palmprint images into differentfrequency
multiresolution subband images for analysis. In decom
posing the image into lower resolution images, WT can
conserve the energy signals and redistribute them into more
compact form. Usually the low frequency subbands contain
most of the energy content and it is able to preserve local
edges well which helps to capture the features that are
insensitive to small distortion. In addition, as the subband
image is only quarter size of the original image, the
computational complexity can be reduced by working on a
lower resolution image. This makes WT distinguishable
from other noise/resolution reduction techniques like spatial
filters with dyadic downsampling.
For readers unfamiliar with PCA, FDA, ICA and wavelet
transformation theories, some brief review of these methods
can be found in Appendix A.
2.3. Feature matching/classification procedures
Identity of an individual can be verified through the
feature matching or classification process. The output of
each feature extraction algorithm produces a feature vector
that is used for classification. The simplest classification
method is based on the concept of similarity where samples
that are similar are classified to the same class. Some
popular similarity measures include the Manhattan (or city
block), Euclidean and Mahalanobis distances. As ICA
produces basis vectors that are not mutually orthogonal, the
cosine distance measure is also employed here. Cosine
measure can be used since ICA allows the basis vectors to
be nonorthogonal, and the angles and distances between
images differ from each other.
Another classification approach is to construct decision
boundaries directly by optimizing an error criterion.
Artificial neural network (ANN) is one such famous
technique. ANN can generalize well on the data it has not
seen before and can take into account the subtle differences
between the modeled data, without the need to assume the
type of relationship and the degree of nonlinearity between
the various independent and dependent variables. In this
research, the Probabilistic neural network (PNN) is
deployed. PNN was first introduced by Specht [19,20] and
it offers several advantages over backpropagation network.
Despite its generalization ability, the training speed of PNN
is much faster because the learning rule is simple and
requires only a single pass through the training data. Most
importantly, PNN new training data can be added anytime
without the need to retrain the entire network [20–22] his is
an important factor in this research when this system is to be
extended to realtime application in the future.
3. Experiment and discussion
3.1. Experiment setup
In our research, a standard PC with Intel Pentium III
processor (1 GHz) and 256 MB random access memories is
used Our input device is the Hewlett–Packard ScanJet
3500c optical scanner. Resolution of 150 dpi, with color
output type in 256 RGB format is adopted when the hand
images are scanned. The original size of the hand image is
about 600!800 pixels but consequently, only the region of
interest (ROI) of the image will be extracted and resized to
150!150 pixels.
The proposed methodology is tested on a modestsized
database containing palm images from 75 individuals.
Thirtyseven of them are females, 46 of them are less than
30 years old, and three of them are more than 50 years old.
The users come from different ethnic groups: 37 Chinese,
followed by 24 Malays, 11 Indian, a Pakistani, an African
and an Iranian. Most of them are students and lecturers from
Multimedia University. To investigate how well the system
can identify unclear or worn palmprints due to labour work,
we have also invited ten cleaners to contribute their
palmprint images to our system.
The users are asked to present their palms at different
directions and stretching degrees when scanned. They do
not need to remove rings or other ornaments from their
fingers when their hand images are taken. The users are
allowed to rotate their palms within G208. If they fail to do
so, an error will be detected in the preprocessing module
and they will be requested to repeat the hand scanning
process again. Fig. 7 illustrates an example of an error
detected during the image acquiring process when the user’s
hand exceeds the permitted angle of rotation.
T. Connie et al. / Image and Vision Computing 23 (2005) 501–515506
Page 7
Each user was requested to provide six images from their
right and left hands with different positions in two
occasions. The average interval between the two occasions
is 21 days. Since the right and left palmprints of each person
are different, both of them are captured and treated as
palmprints from different user. Therefore, there are
altogether 900 (75!6!2) palmprints images in our
database. Among the six images from each palm, three are
selected for training (enrollment) while the other three are
used for testing. Fig. 8 illustrates some palmprint samples in
our database.
3.2. Performance evaluation criteria
The results obtained in this paper are evaluated in terms
of their: (i) correct recognition rate and (ii) verification rate.
Fig. 7. An example of an error detected during the image acquiring process when the user’s hand exceeds the permitted angle of ratation.
Fig. 8. Palmprint samples in the database.
T. Connie et al. / Image and Vision Computing 23 (2005) 501–515 507
Page 8
Correct recognition rate represents the percentage of the
number of people that can be identified by the system. On
the other hand, verification rate is investigated using several
measures like the False Acceptance Rate, False Rejection
Rate, as well as Equal Error Rate. FAR is defined as
FAR ZNumber of accepted imposter claims
Total number of imposter accesses
!100%
(4)
while FRR is defined as
FRR ZNumber of rejected genuine claims
Total number of genuine accesses!100%
(5)
The system threshold value is obtained based on the
Equal Error Rate (EER) criteria where FAR is equals to
FRR. This is based on the rationale that both rates must be as
low as possible for the biometric system to work effectively.
Another performance measurement is obtained from FAR
and FRR which is called Total Success Rate (TSR). It
represents the verification rate of the system and is
calculated as follow:
TSR Z
1K
FACFR
Total number of accesses
??
!100%
(6)
3.2.1. Principal component analysis
In the first experiment, we investigate the performance of
PCA by using different number of principal components (or
feature lengths), varying from 30 to 90. Experimental result
shows that longer feature length leads to higher recognition
rate. Table 1 displays the correct recognition rate of using
the principal components that yield significant changes in
the result. Several classifiers are used to justify the
performance, namely L1and L2distance measure, cosine
measure and Probabilistic Neural Network (PNN).
It is demonstrated by our experiment that as the number
of feature lengths/principal components increases, the
correct recognition rate also increases. The performance
peaks when 55 principal components is used. Itis interesting
to discover that the performance stabilizes, or even begins to
decrease after this point. PNN gives correct recognition rate
of 93.1% onwards after 55 feature lengths, and the other
classifiers indicates the performance is deteriorating.
It can be anticipated that the classification accuracy of
the methods will improve when a more sophisticated
classifier is used. In this research, PNN is used to show
how well the result can improve when a sophisticated
classifier is used.
The verification rates of PCA using the various principal
components are shown in Table 2. We use the distance
measure that maximized the performance, which is L2, for
this purpose.
The palmprint verification method can achieve ideal
result with FARZ2.6% and FRRZ2.6%, respectively.
Fig. 9 shows the Receiver Operating Characteristic (ROC)
curve to serve as a comparison among the performances of
the different principal components.
Based on the experimental result, it can be concluded that
the first few eigenpalms contain the largest variance
direction in the learning set. In this way, we can find
directions in which the learning set has the most significant
amounts of energy. However, as the principal components
increases, it tends to maximize other insignificant infor
mation such as noise, which will decrease the performance
of the system. In fact, it can be shown from the images
generated by using higherorder eigenpalms that they
only contain noise, and do not look like palms at all
Table 1
Correct recognition rates of using different numberof principal components
Number of
feature
length
L1measure
(%)
L2measure
(%)
Cosine
measure
(%)
Probabilis
tic neural
network (%)
30
40
45
50
55
60
70
80
90
89.4
90.2
90.7
90.9
92.4
92.1
91.7
91.1
91.1
90.7
92.2
92.9
92.9
93.1
92.9
92.4
92.4
92.2
85.1
85.7
87.4
89.2
90.4
89.4
89.1
88.4
87.7
91.7
92.4
92.7
93.9
94.1
94.1
94.1
94.1
94.1
Table 2
Performance evaluation of using different principal components
Feature lengthFAR (%)FRR (%)TSR (%) EER (%)
30
40
45
50
55
60
70
80
90
3.3
3.3
3.2
2.6
2.6
3.2
3.3
3.3
3.7
3.3
3.3
3.3
2.6
2.6
3.3
3.3
3.3
4.0
96.7
96.7
96.8
97.3
97.3
96.7
96.7
96.7
96.3
3.3
3.3
3.2
2.6
2.6
3.2
3.3
3.3
3.8
Fig. 9. ROC curve that serves as a comparison among the performance of
the different principal components.
T. Connie et al. / Image and Vision Computing 23 (2005) 501–515508
Page 9
(refer Fig. 10). In this, we deduce that in our palmprint
database, 20% of the principal components are attributed to
true correlation effects while the rest to small trailing
eigenvalues. Therefore, the discarded dimensions are the
ones along which the variance in the data distribution is the
least, which fail to capture enough information for
representation.
3.2.2. Fisher discriminant analysis
To investigate the performance of FDA, the correct
recognition and verification rates are displayed in Tables 3
and 4, respectively. Again, four possible distance metrics
(L1and L2distance measure, Mahalanobis distance and
PNN) are used.
Experimental result shows that FDA is able to achieve
correct recognition rate of 97.7, 95.2 and 95.7% by using
PNN, L1and L2measures, respectively. On the other hand,
verification rate of 98.1% could be attained by using L2
metric.
A comparative experiment betweenFDA/ L2and PCA/L2
has been conducted in this research. The comparison
is made by considering their representative ROC curves.
Fig. 11 shows the ROC curve to compare the performance
of both FDA and PCA.
As expected, experimental result shows that FDA
performs better than PCA. While PCA maximizes all
scatter for data representation, FDA tends to take into
account the within—and betweenclass scatter for classifi
cation. Its intention is to maximize the betweenclass scatter
as to minimize the withinclass scatter.
In choosing the projection that maximizes the total
scatter, PCA retains some unwanted variations [23]. On the
other hand, FDA provides more class separability by
building a decision region between the classes. Therefore,
it is undoubtedly that FDA outperforms PCA as it
transforms the samples into the ‘best separable space’
focusing on the most discriminant feature extraction.
Another reason is that the palmprint images contained in
our database essentially contain lower withinclass variation
as compared to the others. This is due to the use of scanner
based approach in which the images captured are nearly
unaffected by the changes in ambient illumination. There
fore, with this low withinclass variability in our palmprint
nature, and with sufficiently high betweenclass variability,
FDA discriminant ability can be boosted in our experiment.
3.2.3. Independent component analysis
Many criterion functions for ICA were proposed by [24]
based on different search criteria. In this research, the
InfoMax algorithm is deployed. The complete InfoMax
algorithm written in Matlab code is publicly available at:
http://ergo.ucsd.edu/~marni/. In this research, we follow the
experimental setup adopted by Bartlett et al. [25] and set the
parameters for InfoMax as below:
† Block size—50
† Learning rate—The initial learning rate was set to 0.001.
After 1000 iterations, it was reduced to 0.0005, 0.00025
and 0.0001 every 200 epochs subsequently.
† Total number of iterations—1600
Following the discussion in [23], we first apply PCA to
project the data into a subspace of dimension 55 to control
the number of independent components generated by ICA.
The InfoMax algorithm is then applied to the eigenvectors
to minimize the statistical dependence among the resulting
basis images.
Since ICA basis vectors are not mutually orthogonal, the
cosine distance measure is often used to retrieve images in
the ICA subspaces. Tables 5 and 6 present the performance
result reported using cosine, L1and L2measures, as well as
PNN.
Fig. 10. Eigenpalms generated by using different number of feature lengths. (a) Eigenpalms generated using the 55th principal components. (b) Eigenpalms
generated using the 70th principal components. (c) Eigenpalms generated using the 80th principal components. (d) Eigenpalms generated using the 90th
principal components. (e) Eigenpalms generated using the 100th principal components.
Table 3
Correct recognition of FDA using L1 and L2 measures, Mahalanobis
distance and PNN
ClassifiersCorrect recognition rate (%)
L1measure
L2measure
Mahalanobis distance
PNN
95.2
95.7
94.0
97.7
Table 4
Standard error rates of FDA using L2measure
Verification
measures
FARFRRTSR EER
Verification rate (%) 1.92.0 98.11.9
T. Connie et al. / Image and Vision Computing 23 (2005) 501–515509
Page 10
As expected, PNN clearly outperforms the other distance
measures, yielding correct recognition rate of 97%. Among
the other distance measures, cosine measure performs
slightly better than L1 and L2 measures by providing
95.7% correct recognition rate. Subsequently for the
verification rate, 98.0% is achieved using the cosine
measure.
A comparison for ICA/cosine has been made with its
FDA/L1and PCA/L1counterparts by plotting their respect
ive ROCs in Fig. 12.
Our result shows that ICA performs considerably better
than PCA, but does not provide significant advantage over
FDA. Although the idea of computing higher order
moments using ICA is attractive, the assumption that the
palmprint images comprise of a set of independent basis
images is not intuitively clear. Besides, viewing the nature
of palmprint texture that are made up of many crossing and
overlapping ridges, recognition by using localized features
is not suitable. In addition, the fact that some lines are so
thin and unobvious that they will be simply ignored by the
feature extraction algorithm will also confound the
performance of ICA.
To evaluate the computation load taken by the three
methods, the time needed for training and testing (in the
identification stage) 450 templates in our database for each
of these methods are recorded and displayed in Table 7.
According to the recorded measurement, the compu
tation burden required by ICA, approximately half an hour
to compute the basis vector for 450 images, is much greater
than PCA and FDA. The reason lies in the large number of
iterations needed to refine the ICA basis. On the other hand,
there is no significant difference in the time taken by the
algorithms for testing.
In order to reduce the computational time required, we
decided to decompose the images into lower resolution
before further processing. With this, we employ the wavelet
transformation to achieve dimension reduction purpose. The
result of applying wavelet transformation on the images is
provided in the next section.
3.2.4. Wavelet transformation
In feature extraction tasks, the commonly adopted
approach is to select the subband image that contains the
highest energy distribution. Therefore, low frequency
subband is selected upon high frequency subbands for
palmprint structure representation in our research. To justify
our selection, we have adopted three wavelet bases namely
Haar, Daubechies order 2 and Symmlet order 2 (using the
first level decomposition) to demonstrate that the highest
energy distribution is indeed contained in the low frequency
subband and it is able to yield the highest verification rate.
Table 8 illustrates the relationship between the energy
distribution content and verification rate using the three
wavelet bases in our database.
Based on the result shown in the table above, it is obvious
that the high order wavelet basis contains the highest energy
distribution and yields better recognition rate. Our finding is
opposed to the work proposed by Feng et al. [26] which
claimed that highestenergy subband does not necessary
give the best recognition accuracy. In fact our palmprint
Fig. 11. Comparison between PCA and FDA by using ROC curve.
Table 5
Correct recognition rates of ICA
Classifiers Correct recognition rate (%)
L1measure
L2measure
Cosine measure
PNN
95.0
95.0
95.7
97.0
Table 6
Standard error rates of ICA using Cosine measure
Verification
measures
FAR FRR TSREER
Verification rate (%) 2.02.0 98.02.0
Fig. 12. ROC that compares ICA/cosine against FDA/L1and PCA/L1.
T. Connie et al. / Image and Vision Computing 23 (2005) 501–515 510
Page 11
database shows that all the highestenergy subbands
obviously outperform other subband images. Holding this
principle, we focus on the lowfrequency subband for
subsequent analysis.
WefirstintegrateWTwithPCAwiththesystematicuseof
different wavelet families. The integration of WT with PCA
is known as WPCA for brevity. Three decomposition levels
were tested. The first level decomposition decomposes the
imagesfrom150!150 pixelsinto79!79,followedby44!
44 and 26!26 pixels in the first, second and third level
decompositions (as we use the Matlab Wavelet Toolbox, the
decomposed sizes obtained are not half the original image
like in other ordinary cases). Although the image can still be
furtherdecomposedintolowerresolution,westopatthethird
level decomposition. This is based on the rationale that finer
resolution contains less useful information for the recog
nition task. Table 9 presents the performance of using the
different wavelet bases for PCA.
Different wavelet bases exhibit different performances.
We can observe that Daubechies order 5 level 2 yields the
best performance by giving FAR, FRR, TSR and EER of
2.9596, 3, 97.04 and 2.9798%, respectively. In general,
it can be observed that level 3 decomposition performs
poorer than the other two levels of decomposition. This is
due to the reason that the down sampling process eliminates
the change offeature structures of the coarser images, and in
turn causes the discriminant power of the WPCA feature to
be lower than before. This finding confirms our justification
to stop at level 3 decomposition.
Table 7
CPU time used to calculate PCA, FDA and ICA (measured in seconds)
Method PCA (using 55
feature lengths)
FDA ICA
Training time (in second)
Testing time (in second)
72.7
0.4
141.6
0.4
1655.8
0.5
Table 8
Relationship between the energy distributions and verification rates of the
four subband images
Sub band
Haar wavelet basis (level 1)
L1a
D1horizontalb
D1verticalc
D1diagonald
Energy dis
tribution
FAR
FRR
TSR
EER
Daubechies order 2 level 1 wavelet basis
Energy dis
tribution
FAR1.5
FRR1.6
TSR98.4
EER 1.5
Symmlet order 2 level 1 wavelet basis
Energy dis
tribution
FAR 1.5
FRR 1.6
TSR 98.4
EER1.5
99.3480.060 0.0800.003
8.4
9.0
91.6
8.7
24.9
27.0
75.0
25.5
24.1
25.0
75.9
24.5
40.7
41.0
59.2
40.8
99.8300.0620.0580.041
29.4
28.0
70.6
28.7
28.6
30.0
71.4
29.3
46.4
41.0
53.6
43.7
99.8470.0690.077 0.007
28.6
30.0
74.4
29.3
29.4
28.0
70.6
28.7
46.4
41.0
53.6
43.7
aLow frequency subband. High frequency subband details in
bHorizontal.
cVertical.
dDiagonal orientation.
Table 9
Comparative result of using the different wavelet bases on PCA
FilterDecompo
sition level
FAR
(%)
FRR
(%)
TSR
(%)
EER
(%)
Haar1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
2.9
4.1
3.9
2.2
2.9
2.9
2.6
2.9
2.9
2.6
2.9
2.9
2.6
2.9
2.9
2.6
3.9
2.9
2.6
4.0
4.0
2.6
4.0
4.0
2.6
3.0
3.0
2.6
3.0
3.0
2.6
3.0
3.0
2.6
3.0
3.0
2.6
4.0
3.0
2.6
4.0
4.0
97.0
95.9
96.1
97.7
97.0
97.0
97.3
97.0
97.0
97.3
97.0
97.0
97.3
97.0
97.0
97.3
96.0
97.0
97.3
95.9
95.9
2.7
4.0
3.9
2.4
2.9
2.9
2.6
2.9
2.9
2.6
2.9
2.9
2.6
2.9
2.9
2.6
3.9
2.9
2.6
4.0
4.0
Daubechies 4
Daubechies 5
Daubechies 6
Symmlet 6
Symmlet 7
Symmlet 8
Table 10
Wavelet transformation on FDA
FilterDecompo
sition level
FAR
(%)
FRR
(%)
TSR
(%)
EER
(%)
Daubechies 21
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1.4
2.4
4.5
1.3
1.4
4.5
1.5
2.9
3.0
1.3
2.5
3.9
1.4
3.0
4.4
1.4
1.5
2.9
1.3
2.8
4.5
1.3
2.3
4.3
1.4
2.9
4.4
1.4
1.4
4.4
1.4
2.9
2.9
1.4
2.9
4.4
1.4
2.9
4.4
1.4
1.4
2.9
1.4
2.9
4.4
1.4
1.4
4.4
98.5
97.5
95.4
98.6
98.5
95.4
98.4
97.0
96.9
98.6
97.4
96.0
98.5
96.9
95.5
98.5
98.4
97.0
98.6
97.1
95.4
98.6
97.6
95.6
1.4
2.7
4.5
1.4
1.4
4.5
1.5
2.9
3.0
1.4
2.7
4.2
1.4
3.0
4.4
1.4
1.5
2.9
1.4
2.9
4.5
1.4
1.6
4.4
Daubechies 3
Daubechies 4
Symmlet 5
Symmlet 6
Symmlet 7
Symmlet 8
Symmlet 9
T. Connie et al. / Image and Vision Computing 23 (2005) 501–515511
Page 12
Similarly, WT can also be combined with FDA. This
gives rise to another terminology called WFDA. Table 10
shows the performance result by using the different wavelet
bases on FDA.
ExperimentalresultshowsthatDaubechiesorder3Level1,
Symmlet order 8 Level 1 and Symmlet order 9 Level 1 are
most suitable for FDA computation. They yield 1.3, 1.4, 98.6
and 1.4% for FAR, FRR, TST and EER, respectively.
For comparative study, ICA is also combined with
Wavelet transformation, yielding another term called
WICA. The result of WICA is shown in Table 11.
In WICA, Symmlet order 2 Level 1 yields the best result
by providing 1.9, 2, 98 and 1.9% for FAR, FRR, TSR and
EER, respectively.
After testing wavelet with the three subspace projection
techniques, a comparative test is conducted among PCA,
WPCA, FDA, WFDA, ICA and WICA. Fig. 13 depicts this
comparison by using their respective ROC curves.
Based on the diagram above, it can be observed that WT
improves the overall performance of the originaltechniques.
This shows that WT does not only help to reduce the image
size but also increases the performance. WT has indeed
helped to boost the performance of the original method by
proper accounting of global features without loss of
information on key local features. Experimental result
shows that among all the methods, WFDA is able to give the
highest verification rate of 98.6418% when the FAR and
FRR is set low to 1.3569 and 1.4925%, respectively.
4. Conclusion
We have developed a scannerbased palmprint recog
nition system to automatically authenticate the identity of an
individual based on biometric palmprint features. The
proposed system is reliable and user friendly as high
recognition result is provided and convenient acquiring
process is offered.
Our experiments suggest a number of conclusions:
1. Holistic analysis statistical approach is very suitable for
palmprint authentication task. This approach is faster,
less computationally intensive and less prone to
misconceptions in the extraction task used since little a
priori assumptions are made on the nature of the
palmprint.
2. The position and scaling of the palmprint is critical to the
success of palmprint templatebased approach, and the
alignment of the training images is determinant. We
stress that the good performance of the palmprint
recognition method depend on the precision of the pre
processing step.
3. For the feature extraction stage, ICA does not provide
significant advantage over FDA. Thus, it is not
intuitively clear that the palm images comprise of a set
of independent basis images for recognition. It also
suggests that localized feature basis provided by ICA
may not be suitable to represent the crossing and
overlapping ridge structures of the palmprint.
4. The intrinsic structure of the modeled data can boost the
performance of FDA. Images with low withinclass
variability and sufficiently high betweenclass variability
have proven to be able to increase FDA’s performance.
5. As WT conserves energy and redistribute them into more
compact form, performing operations in the wavelet
domain and then reconstructing the result is more
efficient than performing the same operation in the
standard feature space. At the same time, the memory
burden can be reduced.
In this paper, systematic testing and analysis have been
conducted using a modest palmprint database. We are
currently investigating how well the subspace projection
techniques perform when extended to large database.
Table 11
Wavelet transform on ICA
Filter Decompo
sition Level
FAR (%)FRR (%)TSR (%) EER (%)
Symmlet 21
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1.9
2.0
2.0
2.0
1.9
1.9
2.0
1.9
2.0
1.9
1.9
2.0
2.0
2.0
3.1
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
3.0
98.0
98.0
98.0
98.0
98.0
98.0
97.9
98.0
98.0
98.0
98.0
97.9
98.0
98.0
96.8
1.9
2.0
2.0
2.0
1.9
1.9
2.0
1.9
2.0
1.9
1.9
2.0
2.0
2.0
3.0
Symmlet 3
Symmlet 4
Symmlet 5
Symmlet 6
Fig. 13. ROC curve compares the performances among PCA, WPCA, FDA,
WFDA, ICA and WICA, respectively.
T. Connie et al. / Image and Vision Computing 23 (2005) 501–515512
Page 13
We conjecture that exploration in palmprint recognition is
very promising in that it would become an important
complement to the existing biometric technology.
Appendix A. Subspace projection techniques
In this appendix, we provide a brief review of basic PCA,
FDA, ICA and Wavelet Transformation theories. For more
detailed explanation, please refer to [23,24,27] for compre
hensive understanding on the topics.
A.1. Principal component analysis
Let us consider a set of M palmprint images, i1,i2,.,iM,
the average palm of the set is defined as
?i Z1
M
X
M
jZ1
ij
(A.1)
Each palmprint image differs from the average palm?i, by
the vector fnZinK?i. A covariance matrix is constructed
where:
C Z
X
M
jZ1
fjfT
j
(A.2)
Then, eigenvectors, vkand eigenvalues, lkwith sym
metric matrix C are calculated. vkdetermines the linear
combination of M difference images with f to form the
eigen bases:
blZ
X
M
kZ1
vlkfk;
l Z1;.;M (A.3)
From these eigen bases, K(!M) basis are selected to
correspond to the K highest eigenvalues.
The set of palmprint images, is transformed into its eigen
bases components (projected into the palm space) by the
operation:
unkZbkðinK?iÞ
where nZ1,.,M and kZ1,.,K.
The weights obtained form a vector UnZ[un1,un2,.,
unK] that describes the contribution of each eigen basis in
representing the input palm image, treating the eigen bases
as a basis set for the palm images.
(A.4)
A.2. Fisher discriminant analysis
Consider a set of M palmprint images having c classes of
images, with each class containing n set images, i1,i2,.,in.
Let the mean of images in each class and the total mean of
all images be represented by ~ mcand m, respectively, the
images in each class are centered as
fc
nZic
nK ~ mc
(A.5)
and the class mean is centered as
ucZ ~ mcKm(A.6)
The centered images are then combined side by side into
a data matrix. By using this data matrix, an orthonormal
basis U is obtained by calculating the full set of eigenvectors
of the covariance matrix fcT
then projected into this orthonormal basis as follow
nfc
n. The centered images are
^fc
nZUTfc
n
(A.7)
The centered means are also projected into the
orthonormal basis as
^ ucZUTuc
Based on this information, the within class scatter matrix
SWis calculated as
(A.8)
SWZ
X
c
jZ1
X
nj
kZ1
^fj
k^fjT
k
(A.9)
and the between class scatter matrix SBis calculated as
SBZ
X
C
jZ1
nj~ uj~ uT
j
(A.10)
The generalized eigenvectors V and eigenvalues l of the
within class and between class scatter matrix are solved as
follow:
SBV ZlSWV (A.11)
The eigenvectors are sorted according to their associated
eigenvalues. The first cK1 eigenvectors are kept as the
Fisher basis vectors, W. The rotated images, aMwhere aMZ
UTiMare projected into the Fisher basis by
6nkZWTaM
(A.12)
where nZ1,.,M and kZ1,.,MK1.
The weights obtained is used to form a vector YnZ
[6n1,6n2,.,6nK] that describes the contribution of each
fisherpalm in representing the input palm image, by treating
fisherpalms as a basis set for the palm images.
A.3. Independent component analysis
Let s be the vector of unknown source images and x be
the vector of observed mixtures. If A is the unknown mixing
matrix, then the mixing process is written as
x ZA^ s(A.13)
The goal of ICA is to find the separating matrix W such
that
^ s ZWx (A.14)
However, there is no closed form expression to find W.
Instead, many iterative algorithms are used to approximate
W in order to optimize independence of ^ s. Thus, the vector ^ s
is actually an estimate of the true source s. In this research,
T. Connie et al. / Image and Vision Computing 23 (2005) 501–515513
Page 14
the InfoMax principle which was derived from a neural
network perspective is deployed [28].
Sometimes, it is expedient to work on lower dimension
ality. Preprocessing steps can be applied to x to reduce the
dimension space. There are two common preprocessing
steps in ICA. The first step is to centered the images as,
^ x ZxKEfxg
such that Ef^ xgZ0. This enables ICA to deal with only zero
mean data. The next step is to apply whitening transform V
to the data such that
(A.15)
V ZDK1=2RT
(A.16)
where D is the eigenvalues on the diagonal and R is the
orthogonal eigenvectors of the covariance matrix of ^ x. The
whitening process helps to uncorrelate the data so that PCA
can work with unit variance.
In this research, in order to reduce the number of
independent components produced by ICA, PCA is first
appliedtoprojectthedataintoasubspaceofdimensionm,as
described by Bartlett et al. (2002). The InfoMax algorithm is
then applied to the eigenvectors to minimize the statistical
independence among the resulting basis vectors. The pre
application of PCA can discard small trailing eigenvalues
before whitening and reduce computational complexity by
minimizing pairwise dependency [29].
Let the input to ICA, V, be a p by m matrix, where p
represents the number of pixels in the training image, and m
be the first m eigenvectors of a set of n palm images (Section
3.5.1). ICA is performed on VT.
After that, the independent basis vector,^S, is computed
as follows:
^S ZW!VK1
(A.17)
Next, by taking R as the PCA coefficient where RZX!
V, with X representing the n set of zeromean images (image
data is contained in each row), the coefficients matrix of
ICA can be calculated as
B ZR!WK1
(A.18)
Therefore, the reconstruction of the original palmprint
image can be achieved by
X ZB!^S (A.19)
A.4. Wavelet transformation
The wavelet decomposition of a signal f(x) can be
obtained by convolution of signal with a family of real
orthonormal basis, ja,b(x)
ðWjfðxÞÞða;bÞ
ZjajK1=12
ð
R
fðxÞj
xKb
a
??
dxfðxÞ2L2ðRÞ
(A.20)
where a, b2R and as0 are the dilation parameter and the
translation parameter, respectively. The basis function
ja,b(x) are obtained through translation and dilation of a
kernel function j(x) known as mother wavelet as defined
below:
ja;bðxÞ Z2Ka=2jð2KaxKbÞ
The mother wavelet j(x) can be constructed from a
scaling function, f(x). The scaling function f(x) satisfies the
following twoscale difference equation
p X
where h(n) is the impulse response of a discrete filter which
has to meet several conditions for the set of basis wavelet
functions to be orthonormal and unique. The scaling
function f(x) is related to the mother wavelet j(x) via
p X
The coefficients of the filter g(n) are conveniently
extracted from filter h(n) from the following relation
(A.21)
fðxÞ Z
ffiffiffi
2
n
hðnÞfð2xKnÞ
(A.22)
jðxÞ Z
ffiffiffi
2
n
gðnÞfð2xKnÞ
(A.23)
gðnÞ ZðK1Þnhð1KnÞ
The discrete filters h(n) and g(n) are the quadrature
mirror filters (QMF), and can be used to implement a
wavelet transform instead of explicitly using a wavelet
function.
For 2D signal such as images, there exists an algorithm
similar to the onedimensional case for two dimensional
wavelets and scaling functions obtained from onedimen
sional ones by tensiorial product. This kind of two
dimensional wavelet transform leads to a decomposition
of approximation coefficients at level jK1 in four
components: the approximations at level j, and the details
in three orientations (horizontal, vertical and diagonal)
(A.24)
Ljðm;nÞ Z½Hx? ½Hy? LjK1?Y2;1?Y1;2ðm;nÞ
(A.25)
Dj vervitalðm;nÞ Z½Hx? ½Gy? LjK1?Y2;1?Y1;2ðm;nÞ
(A.26)
Dj horizontalðm;nÞ Z½Gx? ½Hy? LjK1?Y2;1?Y1;2ðm;nÞ
(A.27)
Dj diagonalðm;nÞ Z½Gx? ½Gy? LjK1?Y2;1?Y1;2ðm;nÞ
where * denotes the convolution operator, Y2,1 ([2,1)
subsampling along the rows (columns), H and G are a low
pass and bandpass filter, respectively. Similarly, two levels
of the wavelet decomposition is obtained by applying
Wavelet Transform on the lowfrequency band sequentially.
(A.28)
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